Abstract
In this paper, we formulate necessary and sufficient conditions for relative compactness in the space $$BG({\mathbb {R}}_+,E)$$
B
G
(
R
+
,
E
)
of regulated and bounded functions defined on $${\mathbb R}_+$$
R
+
with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space $$BG({\mathbb {R}}_+,E)$$
B
G
(
R
+
,
E
)
. We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps $$BG({\mathbb {R}}_+,E)$$
B
G
(
R
+
,
E
)
into $$BG({\mathbb {R}}_+,E)$$
B
G
(
R
+
,
E
)
and, additionally, be compact.
Solvability of an infinite system of integral equations on the real half-axis
